Predistortion for Nonlinear PA with memory

7/29/2019 Predistortion for Nonli near PA with memory

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PPrreeddiissttoorrttiioonn ffoorr NNoonnlliinneeaarr PPoowweerr

AAmmpplliiffiieerrss wwiitthh MMeemmoorryy

By

Muhammad A. Nizamuddin

Thesis submitted to the faculty of

Virginia Polytechnic Institute and State University

in partial fulfillment of the requirements for the degree of

Master of Science

In

Electrical Engineering

William H. Tranter (chair)

Jeffrey H. Reed

Brian D. Woerner

December 06, 2002

Blacksburg, VA

Keywords: predistortion, nonlinear tapped delay line, Hammerstein systems, indirect

learning

Copyright 2002, Muhammad Nizamuddin


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PPrreeddiissttoorrttiioonn ffoorr NNoonnlliinneeaarr PPoowweerr AAmmpplliiffiieerrss wwiitthh

MMeemmoorryy

By

Muhammad A. Nizamuddin

Committee Chair: William H. Tranter

Electrical Engineering

AABBSSTTRRAACCTT

The fusion of voice and data applications, along with the demand for high data-rate

applications such as video-on-demand, is making radio frequency (RF) spectrum anincreasingly expensive commodity for current and future communications. Although

bandwidth-efficient digital modulation alleviates part of the problem by requiring a

minimal use of spectral resources, they put an extra design burden on RF engineers. RF

transmitters and power amplifiers account for more than half the total maintenance costof a base-station, and occupy nearly the same portion of space. Therefore, power

amplifiers become a bottleneck for digital systems in terms of space and power

consumption. However, power-efficient use of the amplifiers, although desirable, is

extremely detrimental to end-to-end performance due to the very high peak-to-averagepower ratios of modulations that are used today. In order to reduce distortion while

maintaining high power conversion efficiency in a power amplifier, linearization schemesare needed. In addition, significant frequency-dependent Memory Effects result in high

power amplifiers operating on wideband signals. Therefore, these effects need to be

considered during any attempt to minimize amplifier distortion.

In this thesis, we present two schemes to cancel nonlinear distortion of a power amplifier,

along with its memory effects and results for one of the schemes. The results highlightthe fact that in the presence of significant memory effects, cancellation of these effects is

necessary to achieve reasonable improvement in performance through linearization. We

focus on predistortive schemes due to their digital-friendly structure and simpleimplementation. The operating environment consists of a multi-carrier W-CDMA signal.

All of the studies are performed using numerical simulation on MATLAB and AgilentsAdvanced Design System (ADS).


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iii

AACCKKNNOOWWLLEEDDGGEEMMEENNTTSS

First and foremost, all praises are for Him the Beneficent, the Merciful. I would like to

offer my deepest appreciation for my advisor, Dr. Tranter, who has been a constant

source of encouragement. His suggestions and corrections have made a tremendous

difference to my insight into the subject. It was very important to have Philip Balister, my

project-mate, each step of the way for discussing ideas and implementations, and I am

grateful for his insights and interest. My thanks to Dr. Reed, who worked closely with us

and has played a pivotal role in initiating this research effort. Dr. Bostian of the Center

for Wireless Telecommunication of Virginia Tech kindly provided us with the ADSamplifier model which was used for all the studies, and I acknowledge that assistance. I

cannot forget all my colleagues at the Mobile and Portable Radio Research Group,

especially Farhan, Yasir, Sarfraz, and Patrick, who listened to my incessant explanations

and, unknowingly at times, gave useful suggestions and support. My sincere appreciation

is due to Dr. Efstathiou and all the engineers at the Digital Radio Group, Analog Devices

Inc. They have shared a significant amount of their expertise in WCDMA systems, with

useful insights and suggestions during our meetings. I would also like to thank Analog

Devices Inc. for funding this project.

My parents and siblings have, and continue to offer, tremendous support for my higher

studies and I am indebted to them and their prayers. It is their belief in me that has made

me work harder and achieve more.


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iv

TTAABBLLEE OOFFCCOONNTTEENNTTSS

1 - INTRODUCTION................................................................................................................................... 1

1.1 MOTIVATION FOR THE RESEARCH ........................................................................................................ 1

1.1.1 Why Linear Amplifiers? ............................................................................................................... 2

1.1.2 Tradeoff between Linearity and Efficiency in a PA ..................................................................... 2

1.1.3 Linearization of a Nonlinear Amplifier........................................................................................ 2

1.1.4 Modeling and Simulation of RF Power Amplifiers ...................................................................... 4

1.2 OUTLINE OF THE THESIS ....................................................................................................................... 5

2 - DISTORTION ANALYSIS AND MODELING OF A POWER AMPLIFIER................................. 6

2.1 INTRODUCTION..................................................................................................................................... 6

2.2 NONLINEARPOWERAMPLIFIER............................................................................................................ 6

2.3 NARROWBAND DISTORTION ................................................................................................................. 8

2.3.1 AM-AM Distortion ..................................................................................................................... 11

2.3.1.1 Two-tone power series analysis........................................................................................................... 11

2.3.1.2 Two-tone envelope analysis ................................................................................................................ 13

2.3.2 AM-PM Distortion ..................................................................................................................... 14

2.4 WIDEBAND DISTORTION..................................................................................................................... 15

2.4.1 Memory Effects .......................................................................................................................... 17

2.4.1.1 Electrical memory effects.................................................................................................................... 18

2.4.1.2 Thermal memory effects ..................................................................................................................... 20

2.4.2 Characterization of Memory Effects .......................................................................................... 22

2.5 BEHAVIORAL MODELING OF POWERAMPLIFIERS .............................................................................. 23

2.5.1 Envelope Simulation of Power Amplifiers ................................................................................. 23

2.5.2 Narrowband Amplifier Model.................................................................................................... 24

2.5.3 Models with Memory.................................................................................................................. 25

2.5.3.1 Volterra models................................................................................................................................... 25

2.5.3.2 Wiener models .................................................................................................................................... 26

2.5.3.3 Nonlinear Tapped delay line (TDL) models........................................................................................ 26

2.5.4 Modeling Results for a Power Amplifier with Memory.............................................................. 27

2.5.4.1 Data collection setup ...........................................................................................................................282.5.4.2 Circuit Modeled and TDL-Modeled envelope characteristics............................................................. 28

2.6 SUMMARY ......................................................................................................................................... 29


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3 - POWER AMPLIFIER LINEARIZATION ........................................................................................ 34

3.1 INTRODUCTION................................................................................................................................... 34

3.2 CANCELINGNONLINEARDISTORTION................................................................................................ 35

3.2.1 Cartesian Feedback Method...................................................................................................... 35

3.2.2 Feedforward Amplifier............................................................................................................... 36

3.2.3 Predistortion.............................................................................................................................. 39

3.2.3.1 Analog predistorter.............................................................................................................................. 41

3.2.3.2 Digital predistortion ............................................................................................................................ 41

3.3 LINEARTRANSMITTERS...................................................................................................................... 44

3.3.1 Linear Amplification using Nonlinear Components (LINC) ...................................................... 44

3.3.2 Envelope Elimination and Restoration (EER)............................................................................ 47

3.4 SUMMARY .......................................................................................................................................... 49

4 - PREDISTORTION WITH MEMORY CANCELLATION.............................................................. 50

4.1 INTRODUCTION................................................................................................................................... 50

4.2 HAMMERSTEIN SYSTEM ..................................................................................................................... 50

4.2.1 Model-based Learning Scheme.................................................................................................. 51

4.2.1.1 Least Squares Estimation (LSE) in complex domain .......................................................................... 53

4.2.1.2 Solution to the Hammerstein Problem................................................................................................. 54

4.2.2 Adaptation using Indirect Learning........................................................................................... 55

4.2.3 Disadvantages of Hammerstein System ..................................................................................... 58

4.3 TAPPED-DELAY-LINE (TDL) PREDISTORTION ................................................................................... 59

4.3.1 Adaptation Procedure................................................................................................................ 604.4 MATLABSIMULATION RESULTS FORTDL PD............................................................................... 61

4.4.1 Spectral Improvement................................................................................................................ 62

4.4.2 Mean Squared Envelope Error Analysis.................................................................................... 67

4.5 SIMULATIONS ON ENVELOPE SIMULATOR IN ADS ............................................................................. 69

4.5.1 Envelope Simulator Description ................................................................................................ 70

4.5.1.1 Predistorter in ADS .............................................................................................................................71

4.5.2 Results........................................................................................................................................ 74

4.5.2.1 Comparison between Memory less and TDL Predistortion................................................................. 74

4.5.2.2 Sampling rate studies ..........................................................................................................................774.5.2.3 Comments ........................................................................................................................................... 78

4.6 SUMMARY .......................................................................................................................................... 80


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5 - SUMMARY AND DISCUSSION......................................................................................................... 81

5.1 BRIEF SUMMARY ................................................................................................................................ 82

5.2 FUTURE WORK................................................................................................................................... 83

5.3 CONCLUDING REMARKS ..................................................................................................................... 85

APPENDIX A ............................................................................................................................................. 86

APPENDIX B.............................................................................................................................................. 88

BIBLIOGRAPHY ...................................................................................................................................... 90

VITA............................................................................................................................................................ 95


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TTAABBLLEE OOFFFFIIGGUURREESS

FIGURE 1.1 EFFICIENCY VS. INPUT BACKOFF FOR VARIOUS POWER AMPLIFICATION METHODS (A AND B

REFER TO CLASS-A, AND CLASS-B OPERATION.) [RAAB02]................................................................... 3

FIGURE 2.1 SPECTRAL REGROWTH IN A SINGLE-CARRIERWCDMA ............................................................ 7

FIGURE 2.2 SIGNAL CONSTELLATION FOR A 16-QAM SIGNAL AT THE OUTPUT OF A NONLINEAR AMPLIFIER

SHOWING PHASE ROTATION AND AMPLITUDE COMPRESSION.................................................................. 7

FIGURE 2.3 A MODELED AM-AM ENVELOPE TRANSFER CHARACTERISTICS OF A PA (P1DB IS THE 1-DB

COMPRESSION POINT)............................................................................................................................. 9

FIGURE 2.4 - AN FSKSIGNAL HAVING 1 BIT PER SYMBOL. ............................................................................. 9

FIGURE 2.5 - AN RRC-FILTERED, 16-QAM ENVELOPE HAVING 3 BITS PER SYMBOL. ................................... 10

FIGURE 2.6 - BANDPASS REPRESENTATION OF A POWER AMPLIFIER. ............................................................. 14

FIGURE 2.7 - TWO BASIC AMPLIFICATION SCHEMES USED IN MULTI-CARRIER TRANSMITTERS (RFPC DENOTES

RF POWER COMBINATION AND BBPC STANDS FOR BASEBAND POWER COMBINATION)....................... 16

FIGURE 2.8 - AM-AM CURVE FOR A 4-CARRIERW-CDMA SIGNAL SHOWING HYSTERISIS. ......................... 16

FIGURE 2.9 - MEASURED AND MODELED PHASE OF IM3 AS A FUNCTION OF TONE SPACING [VUOLEVI01].... 18

FIGURE 2.10 SCHEMATIC SHOWING THE INTERACTION OF THE TWO NONLINEARITIES [VUOLEVI00].......... 19

FIGURE 2.11 - THE GENERATION OF IM3 FROM SECONDARY MECHANISMS [VUOLEVI01]. ........................... 20

FIGURE 2.12 THERMAL MODEL OF A BJT IN (A) PHYSICAL AND (B) ELECTRICAL CONFIGURATION, SHOWING

TEMPERATURE AT VARIOUS LEVELS [VUOLEVI01]............................................................................... 21

FIGURE 2.13 - MEASUREMENT SETUP FOR CHARACTERIZING 3G AMPLIFIERS . ............................................. 23

FIGURE 2.14 - THE WIENER STRUCTURE FOR NONLINEAR SYSTEMS WITH MEMORY. .................................... 26

FIGURE 2.15 - TIME-DELAY LINE MODEL FOR A PA....................................................................................... 27

FIGURE 2.16 SNAPSHOT OF THE DATA GENERATING PORTION OF THE ENVELOPE SIMULATOR.................... 30

FIGURE 2.17 PA CIRCUIT MODEL SECTION.OF THE ENVELOPE SIMULATOR................................................. 31

FIGURE 2.18 COMPARISON BETWEEN OUTPUT ENVELOPE FOR4-CARRIERWCDMA. ................................ 32

FIGURE 2.19 - MEASURED AND MODELED AM-AM CHARACTERISTICS (POLAR METHOD REFERS TO THE

MEMORYLESS POLYNOMIAL METHOD). ................................................................................................ 32

FIGURE 2.20 - PSD COMPARISON AT THE OUTPUT OF THE PA FOR A 4-CARRIERWCDMA. ......................... 33

FIGURE 3.1 - CARTESIAN FEEDBACK TRANSMITTER. ..................................................................................... 35FIGURE 3.2 - BLOCK DIAGRAM OF A FEEDFORWARD SYSTEM ....................................................................... 37

FIGURE 3.3 A BASIC PREDISTORTION TRANSMITTER. ................................................................................. 39

FIGURE 3.4 - RELATIONSHIP BETWEEN RLO AND LINEAR GAIN OF A GENERAL PREDISTORTED SYSTEM.

(RANGE-1 AND RANGE-2 REPRESENT RLO FOR CURVE-1 AND -2, RESPECTIVELY.)............................ 40

FIGURE 3.5 - COMPLEX GAIN BASED PREDISTORTER..................................................................................... 42

FIGURE 3.6 - A HAMMERSTEIN-WIENER SYSTEM PAIR. ................................................................................ 44


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FIGURE 3.7 BASIC OUTLINE OF A LINC TRANSMITTER. ............................................................................. 45

FIGURE 3.8 - SPLITTING OF A NON-CONSTANT ENVELOPE SIGNAL, INTO TWO CONSTANT-ENVELOPED PM

SIGNALS. .............................................................................................................................................. 47

FIGURE 3.9 SIMPLIFIED BLOCK DIAGRAM OF KAHNS EERTECHNIQUE. .................................................... 48

FIGURE 4.1 A HAMMERSTEIN-WIENER SYSTEM......................................................................................... 52

FIGURE 4.2 - NONLINEARITY ADAPTATION FOR THE HAMMERSTEIN SYSTEM. .............................................. 52

FIGURE 4.3 - PRE-EQUALIZATION FORPA MEMORY EFFECTS. ...................................................................... 52

FIGURE 4.4 - INDIRECT LEARNING SCHEME FOR A HAMMERSTEIN-TYPE PREDISTORTER. .............................. 56

FIGURE 4.5 A GENERALIZED 3-TAP, DELAY LINE STRUCTURE. ................................................................... 59

FIGURE 4.6 BLOCK DIAGRAM OF THE ADAPTATION SCHEME FOR NONLINEARTDL PD. ............................ 61

FIGURE 4.7 SPECTRAL COMPARISON FOR A SINGLE CARRIERW-CDMA. THE TDL PD HAS A DELAY

LENGTH OF 3, AND A 6TH

ORDER POLYNOMIAL AT EACH TAP. ............................................................... 62

FIGURE 4.8 SPECTRAL COMPARISON FOR A TWO- CARRIERW-CDMA. THE TDL PD HAS A DELAY LENGTH

OF 3, AND A 6

TH

ORDER POLYNOMIAL AT EACH TAP. ............................................................................ 63FIGURE 4.9 SPECTRAL COMPARISON FOR A THREE-CARRIERW-CDMA. THE TDL PD HAS A DELAY

LENGTH OF 3, AND A 6TH ORDER POLYNOMIAL AT EACH TAP. ............................................................... 63

FIGURE 4.10 SPECTRAL COMPARISON FOR A FOUR-CARRIERW-CDMA. THE TDL PD HAS A DELAY

LENGTH OF 3, AND A 6TH

ORDER POLYNOMIAL AT EACH TAP. ............................................................... 64

FIGURE 4.11 COMPARISON BETWEEN EPSD FOR A SINGLE-CARRIERW-CDMA. ...................................... 65

FIGURE 4.12 - COMPARISON BETWEEN EPSD FOR A 2-CARRIERW-CDMA. ................................................ 65



FIGURE 4.15 NMSE VS. THE LENGTH OF TDL PD WITH A 6TH

ORDER POLYNOMIAL AT EACH TAP FOR

DIFFERENT NUMBER OF WCDMA CARRIERS. ...................................................................................... 67

FIGURE 4.16 - NMSE VS. POLYNOMIAL ORDER AT EACH TAP, FOR A 4-TAP TDL PD, WITH DIFFERENT

NUMBER OF CARRIERS.......................................................................................................................... 68

FIGURE 4.17 - NMSE VS. INPUT BACKOFF FOR A 3-CARRIERW-CDMA OPERATING ON A 4-TAP, 6TH ORDER

TDL PD............................................................................................................................................... 69

FIGURE 4.18 - THE PREDISTORTER BLOCK DESIGNED FORADS SIMULATIONS. ............................................ 71

FIGURE 4.19 - SCHEMATIC OF THE ADS ENVELOPE SIMULATOR USED TO PERFORM SIMULATIONS FOR

DIFFERENT BASEBAND SAMPLING RATES. ............................................................................................ 72

FIGURE 4.20 - FRAME STRUCTURE OF A DOWNLINKDPCH (SOURCE [3GPP02]). ........................................ 73FIGURE 4.21 - MAGNITUDE OF THE SIGNAL ENVELOPE FOR A 2-CARRIERW-CDMA CARRYING ONE DPCH

FRAME. ................................................................................................................................................ 73

FIGURE 4.22 MEASURED AND MODELED AM-AM CHARACTERISTICS FOR THE PA OPERATING ON A 2-

CARRIERW-CDMA CARRYING ONE DPCH FRAME............................................................................. 74

FIGURE 4.23 - COMPARISON BETWEEN MEASURED AM-AM FOR A 2-CARRIERW-CDMA........................... 75


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FIGURE 4.24 - NMSE COMPARISON FOR MEMORYLESS AND MEMORY PREDISTORTION. ............................... 75

FIGURE 4.25 - NPSD PLOTS COMPARING MEMORYLESS AND TDL PREDISTORTION SCHEMES, ..................... 76


FIGURE 4.27 - MEAN PHASE SHIFT AT THE OUTPUT OF PA VS. INPUT BINS. ................................................... 77

FIGURE 4.28 NPSD PLOTS FOR STUDYING CANCELLATION PERFORMANCE VS. THE BASEBAND SAMPLING

FREQUENCY FOR A 2-CARRIERW-CDMA INPUT AND A 2-TAP, 5TH ORDERTDL PD............................ 79

FIGURE 4.29 - NMSE PLOTS FOR STUDYING CANCELLATION PERFORMANCE VS. THE BASEBAND SAMPLING

FREQUENCY FOR A 2-CARRIERW-CDMA AND A 2-TAP, 5TH ORDERTDL PD. ................................... 79

FIGURE 4.30 - MEAN PHASE SHIFT AT THE OUTPUT OF PA VS. INPUT BINS. ................................................... 80


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11 -- IInnttrroodduuccttiioonn

This is the wireless era. Growth of the wireless industry has been tremendous since the

beginning of last decade, while the thirst for higher data rates has been unquenchable.

The proliferation of service providers as a result of this extensive growth, along with the

integration of voice and data services, has made the radio frequency (RF) spectrum a rare

and valuable commodity. The important question for all commercial communication

system manufacturers is How do we squeeze as much data as possible in a given portion

of RF spectrum? The answer to this question has led researchers to digital modulation

schemes that are more bandwidth-efficient, i.e., carry more bits per hertz when compared

with a narrowband modulation. This enhancement, however, comes at a cost. The rapidly

time-varying envelope of the resultant modulation is unsuitable for a nonlinear RF

transmitter, or in particular, a power amplifier (PA).

1.1 Motivation for the Research

Power amplifiers have nonlinear characteristics in the most efficient region of operation.

Any signal whose envelope is fluctuating inside this region is severely distorted

depending on the variance of those fluctuations. The critical trade-off RF engineers have

to make is between linearity and efficiency of a PA and, not surprisingly, the trade-off

becomes increasingly complex as the modulation bandwidth increases.

This thesis attempts to present the various distortion phenomena occurring in a power

amplifier and the impact of those on WCDMA, which is a part of the 3G standard,

promising increased capacities and improved reliability in a very dynamic, wireless

channel.


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1.1.1 Why Linear Amplifiers?

The linearity requirement in a PA is based on the following simple rule: A linear

modulation should be processed by linear blocks to limit distortion. With a nonlinear

transmitter, the fluctuating envelopes of linear modulation schemes receive non-uniform

amplification within the compression region, generating inter-modulation (IM) products.

These products cause out-of-band emissions and in-band distortion, and are known as

Inter Modulation Distortion (IMD). Out-of-band emissions, orspectral regrowth, result

in an increased transmission bandwidth, and act as interference in adjacent channels, thus

degrading system level performance. In-band distortion causes self-interference, which

appears as distortion in signal constellation and increased bit-error rates at the receiver.

1.1.2 Tradeoff between Linearity and Efficiency in a PA

In a transmitter, a high value of power conversion efficiency is critical for effective

resource utilization. Increasing this efficiency lowers the cost per output watt. The power

conversion efficiency is defined in several ways, but the most common is the ratio of

output power to the power required for DC bias. Thus, it is obvious that for a fixed DC

bias, efficiency increases as the operating point moves closer to the saturation point that

exists in the severely nonlinear region of operation. For a constant-envelope modulation

this is easily achieved without compromising linearity, but distortion is inevitable when

the input signal envelope fluctuates around the operating point.

1.1.3 Linearization of a Nonlinear Amplifier

The efforts to limit the amount of distortion, introduced by an amplifier have been

continuing since the advent of power amplification theory itself. The simplest scheme for

limiting distortion is to operate a PA in its linear region using power back off, or down-

scaling of the input signal power. However, the linear region of the PA has a very poorpower conversion efficiency (Figure 1.1). As a result, amplifiers with very high peak-to-

average output power ratio have to be used, which results in increased maintenance cost

and requires more space; a bottleneck, as the number of base stations, per square mile


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continues to increase. The battery life for handsets1

is also reduced with such a power

backoff scheme, which is unacceptable for the wireless industry as it works on

introducing more features in smaller portable units requiring longer battery lives. Thus,

power back off is not a solution for most transmitters today, and amplifier linearization is

the only feasible option for the designer.

Figure 1.1 Efficiency vs. Input backoff for various power amplification methods (A andB refer to class-A, and class-B operation.)

(Source: [Raab02], Copyright IEEE, 2002. Used with permission.)

Linearization schemes process the signal in a transmitter chain so that the signal at the

output of the transmitter is a linearly amplified version of its input. All linearizers use

amplitude and phase of the input as a reference for comparison with the output, and use

the difference in some form to cancel the IMD. There are several ways that this can be

achieved; complexity, bandwidth of operation, performance, and implementation issues

vary. Feedback schemes, Feedforward amplifiers, and predistortion, are common. Other

schemes are being re-introduced, which use amplifiers in different configurations to

produce linearly-amplified versions of the input. Examples of these schemes include

1 The terms handsets, user terminals, portable units, and cell phones refer to the same thing in this thesis.


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Envelope Elimination and Restoration (EER) and Linear amplification using Nonlinear

Components (LINC). Not all linearization schemes are suitable for a digital

implementation, which is convenient with the growth of digital signal processing

technology and software-defined radio architectures.

This research effort is focused on predistortive linearizers in the digital domain. This

form of linearization has distinct advantages with current digital reconfigurable

architectures. We have focused on adaptive schemes, which are better suited for tracking

drifts in the PA parameters and characteristics.

1.1.4 Modeling and Simulation of RF Power Amplifiers

The subject of numerical simulation and its advantages has been discussed extensively in

the literature [Jeruchim92, Schneider93, Tranter88, and Tranter92]. Numerical

simulations are almost unavoidable when doing accurate studies on nonlinear systems,

due to the non-tractable nature of their mathematical analysis. These numerical

simulations require a suitable model for the system under study.

Nonlinear amplifiers have, traditionally, been modeled using mathematical functions

characterizing their measured amplitude (AM-AM) and phase (AM-PM) nonlinearities.

These traditional memoryless amplifier models, however, do not take the memory effectsinto account, which are visible in high power amplifiers operating on wideband signals.

In order to perform the numerical simulation of amplifiers exhibiting nonlinearity and

memory effects, there is a need for amplifier models with memory. The accuracy of a

simulation is critically tied to the accuracy of the numerical models for the components.

Therefore, it is necessary to spend considerable effort in devising accurate models for

power amplifiers to conduct useful numerical studies on them. It should not come as a

surprise however, that simulation-based study of an RF subsystem is not sufficient for

their complete understanding due to their very complex nonlinear physical nature.

Therefore, the subject of time-efficient numerical simulation of a nonlinear device is still

in its infancy and an active area in research.


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1.2 Outline of the Thesis

The rest of the thesis has the following order. Chapter 2 presents the motivation and

ground-work for the research. A brief discussion of memory effects is included, and some

of the models developed for modeling these effects are outlined.2 Chapter 3 presents a

literature study on common classes of linearization and linear transmitters. Emphasis has

been put on predistortion-based linearization. Chapter 4 gives a detailed study of model-

based and model-less predistortion methods with memory cancellation. We have

concentrated on Tapped delay line (TDL) schemes, but have included a description of

Hammerstein systems. Algorithms are developed for the Hammerstein and TDL models

and results are presented for the TDL scheme, given its several advantages over the

Hammerstein system. The results are compared with traditional memoryless predistortion

schemes. Chapter 5 concludes the thesis with recommendations for future research in this

direction.

2 The detailed analysis and development of numerical models for power amplifiers with memory was not a

part of the research effort conducted by the author.


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22 -- DDiissttoorrttiioonn AAnnaallyyssiiss aanndd MMooddeelliinngg ooffaa

PPoowweerr AAmmpplliiffiieerr

2.1 Introduction

Any system, mechanical, electrical, or electro-mechanical, has two basic regions of

operation: Desired and Undesired. In order to optimally use a system, careful

understanding of its behavior during optimal and suboptimal operation is critical.

Likewise, proper understanding of the working of a power amplifier, along with the

distortion introduced during its operation, is necessary for maximizing efficiency while

keeping distortion at a minimum.

This chapter begins with a discussion on the causes of distortion in a PA, including

memory effects. Conventional modeling techniques for narrowband power amplifiers

have been included before a brief discussion on modeling procedures for a PA with

memory, along with the results of model estimation procedure.

2.2 Nonlinear Power Amplifier

Wherever there is wireless communication, there are transmitters, and wherever there are

transmitters, there are RF power amplifiers (PA) [Cripps99]. Ideally, a power amplifier is

required to amplify signals, before they are sent out on airwaves in a way that results in a

constant gain over the entire dynamic range, or frequency spectrum, of the input.

Unfortunately, this is not possible for input signal levels approaching the rated value of a

PA, since it starts to saturate, resulting in gain compression or reduction in available gain.

In addition, the input signal experiences phase variations near the compression region,

causing synchronization error for analog modulations, or additional skewing of signal

constellation in case of a digital modulation. A common artifact of amplifier distortion is

spectral regrowth in the output spectrum (Figure 2.1). Skewing and compression of the


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signal constellation, in the case of digital modulation, is another undesired effect, causing

timing synchronization errors at the receiver. Figure 2.2 shows signs of amplifier

distortion on the signal constellation of a 16-QAM signal.

Figure 2.1 Spectral regrowth in a single-carrier WCDMA

-500 -400 -300 -200 -100 0 100 200 300 400 500-500

-400

-300

-200

-100

0

100

200

300

400

500

I-channel

Q-channel

Figure 2.2 Signal constellation for a 16-QAM signal at the output of a nonlinearamplifier showing phase rotation and amplitude compression.


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2.3 Narrowband Distortion

A power amplifier is benign, as long as it is operated in its linear region. However, when

a power amplifier is operated close to the compression region using a time-varying signal

envelope, it generates unwanted by-products. In order to develop a basic understanding

of the generation process for these distortion products, mathematical analysis using

suitable models is necessary.

In this section, our purpose is not to explain a subject that is widely covered in many texts

such as [Cripps99], [Kenington00], or [Tranter02] but to provide a useful overview in

order to justify the need for the research outlined in this thesis.

In order to develop a basic understanding for distortion in a nonlinear amplifier, we beginfrom the mathematical representation of a general nonlinearity. The most common

representation is a polynomial or power series such as

2 3

1 2 3y a x a x a x= + + (2.1)

Here, 1a x is the desired output, and the rest of the terms are by-products. The number

of terms after the linear component 1a x , depends on the position of the operating point

and the severity of the compression region. Figure 2.3 outlines the linear and

compression region of a typical amplifier. Modulations like FM, FSK, and PSK, have

constant envelopes (Figure 2.4). Thus, their operating points can be set deep in the

compression region, resulting in power efficient and distortion-less operation. Therefore

these modulations are suitable for applications requiring high reliability, such as military

communications. In commercial applications, however, they have been replaced with

bandwidth efficient modulations such as QAM for carrying high data rates using minimal

spectral resources.


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Figure 2.3 A modeled AM-AM envelope transfer characteristics of a PA (P1dB is the

1-dB compression point).

Figure 2.4 - An FSK signal having 1 bit per symbol.

Linear

Region

Compression

RegionP1dB


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Figure 2.5 illustrates the envelope of a QAM signal with a peak-to-average-power ratio of

6.3 dB, which is severely distorted if PA is operated close to P1dB. Therefore, for

minimal distortion the PA is operated deep in its linear region resulting in inefficient

operation.

Figure 2.5 - An RRC-filtered, 16-QAM envelope having 3 bits per symbol.

When the theory of PA distortion was developed the modulations were narrowband, or

satisfied the narrowband assumption, which states that a signal is narrowband compared

with the bandwidth of an amplifier when (a) the carrier frequency is much larger than

the signal bandwidth, and more importantly, (b) the amplifier gain is flat over the entire

bandwidth of the signal. When a PA is operated with such a signal, the signal undergoes

two mechanisms: An AM-AM mechanism translates the input voltage (power) into an

output voltage (power), and an AM-PM mechanism, where the PA introduces a phase

shift to the input depending on the instantaneous voltage (power) of the input. Since this

characterization of the PA is based on the narrowband assumption of the input signal, the


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resultant distortion is called Narrowband Distortion. Below we analyze the two distortion

mechanisms in detail.

2.3.1 AM-AM Distortion

AM-AM in a PA occurs when the amplitude modulation of a signal is converted into

another amplitude modulation accompanied by power amplification of the signal. If the

PA is linear, AM-AM is just an amplitude-scaling operation. However, when the signals

amplitude modulation, or envelope, has peaks in the compression region, the AM-AM

process becomes nonlinear and introduces distortion. The resultant distortion is called

AM-AM distortion. The simplest way to analyze this distortion mathematically is through

a two-tone test.

2.3.1.1 Two-tone power series analysis

A two-tone input with equal amplitudes is of the form,

1 2cos cosx v t v t = + .

When this signal is passed through the third order nonlinearity given in (2.1) the output is

2

1 1 2 2 1 2

3

3 1 2

(cos cos ) ( cos cos )( cos cos )

y a v t t a v t v ta v t v t

= + + +

+ +(2.2)

( )3

2 32 1 1 2

2

2 1 2 1 2 1 2

1 2

3

3 1 2 1 2

2 1 2 1

9cos cos

4

1 1cos 2 cos 2 cos( ) cos( )

2 2

1 1cos3 cos3

4 4

3 3cos(2 ) cos(2 )

4 4

3 3cos(2 ) cos(2 )

4 4

a vy a v a v t t

a v t t

t t

a v t t

t t

= + + +

+ + + + +

+

+ + + + + + +

(2.3)


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12

According to (2.3), small amplitudes result in negligible distortion. However, as the input

drive level surges, we can observe an increase in the strength of the distortion products.

The second term in (2.3) shows a nonlinear increase in the amplitude of the fundamental

frequency components. The cross-modulation terms, or terms carrying the various sums

and differences of input harmonics, are collectively known as Inter-Modulation

Distortion or IMD, since they smear or distort the original input spectrum. Two kinds of

distortion terms can be observed in (2.3). The even order terms, or the terms having an

even sum of harmonics, fall outside the band of interest, and are collectively termed as

out-of-band distortion, while the odd order IM products that interfere with the

frequencies of interest constitute the in-band distortion. Out-of-band spectral components

are collectively termed as spectral regrowth, and quantified using Adjacent channel

Leakage power Ratio (ACLR or ACPR). ACLR is commonly used to specify linearity

specifications in terms of the allowable spectral regrowth. It is defined as the ratio of the

RRC-filtered mean power, centered on the assigned channel frequency, to the RRC-

filtered mean power, centered on an adjacent channel frequency [3GPP02-03]. Strict

limits on ACLR values are imposed by the Federal Communications Commission (FCC)

for limiting adjacent channel interference. Table 1 shows the ACLR requirement at the

output of a Base Station (BS) transmitter operating on a W-CDMA signal [3GPP02].

BS channel offset below the first orabove the last carrier frequency used

ACLR limit

5MHz 45 dB

10MHz 50 dB

Table 1 - Minimum ACLR requirements for a W-CDMA base station.

The in-band distortion increases frame-error and bit-error rates, thus affecting end-to-end

performance. In this context, Total Harmonic Distortion, or THD, is commonly used to


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classify amplifiers with respect to their nonlinear behavior, defined as the ratio of power

in all frequencies of the output signal other than the fundamental to the power in

fundamental frequencies.

Third-order terms severely degrade the end-to-end performance, since they have strong

spectral components interfering with the desired band. However, as the nonlinearity

becomes stronger, the effects of fifth and higher order terms become equally important.

In (2.3) frequency components at 1 22 , 1 22 + , 2 12 and 2 12 + are third-

order components. 1 22 and 2 12 are referred to as IM3 components. The 2nd

order harmonics 12 and 22 generate additional IM3 components through secondary

nonlinear mechanisms. Such mechanisms result in frequency-dependency of the envelope

characteristics of a PA [Section 2.4.1.1].

2.3.1.2 Two-tone envelope analysis

In order to describe the distortion in terms of the signal envelope, a slight modification in

the mathematical representation of our two-tone signal is needed. The two-tone signal is

1 2cos cosv t v t = +

The above expression can be written as

1 2 1 22 cos cos2 2

x v t t +

=

Now defining, 1 2 1 2and2 2

m

+ = = we have,

2 cos cosmv t t = (2.4)

Equation (2.4) represents a double sideband, suppressed carrier, AM-modulated signal,

with a PAPR of 3 dB. The signal at the output of an amplifier, modeled as a polynomial

nonlinearity, can be represented in the form [Cripps99],


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[ ]

[ ]

11 12 13

21 22 23

cos cos 2 cos3 cos

cos cos 2 cos3 cos 2

m m m

m m m

y A t t t t

B t t t t

= + + + +

+ + + +

"

. (2.5)

In the above expression, the various harmonics of input, generated through the nonlinear

process, interact with the harmonics of the carrier to generate IMD. If we assume that the

modulating signals bandwidth is sufficiently small compared to the carrier frequency,

then the higher distortion products, resulting from the interaction between input

harmonics and higher carrier harmonics, are too far away from the distortion products

around the fundamental carrier frequency, and are assumed to be completely filtered off

using a zonal filter (Figure 2.6). In (2.5), the IM3 components are generated through the

interaction between the 3 m term and the fundamental harmonic of carrier.

Figure 2.6 - Bandpass representation of a power amplifier.

2.3.2 AM-PM Distortion

AM-AM distortion is a properly-understood subject, and can be directly tied to gain

compression of a nonlinear amplifier. Phase nonlinearity, however, is not as well-

understood. The base-collector capacitance in a BJT is a critical source of amplitude-

dependent phase shifts, whereas the depletion and junction nonlinear resistances

introduce phase variations in an FET due to their dependency on the input signal level

[Cripps99]. In any case, these level-dependent phase variations result in additional IM3

products and rotation of the signal constellation in the case of digital modulation (Figure


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15

2.2). The phase nonlinearity is also termed as quasi-static memory of the amplifier, since

it depends on the instantaneous input, but introduces level-dependent delay at the output.

2.4 Wideband Distortion

In order to make efficient use of resources, a combination of multiple access techniques

is typically used. One standard feature in 3G is to use CDMA with Frequency Division

Duplex (FDD) or Time Division Duplex (TDD). In the FDD mode, individual WCDMA

signals are modulated on separate carrier frequencies. There are two basic techniques for

the transmission of these carriers as outlined in Figure 2.7. The conventional setup uses a

transmitter chain for each carrier having a single carrier power amplifier (SCPA) and a

tunable cavity filter for limiting interference between carriers. The output signals are

added together in an RF power combiner (RFPC) before being fed to the antenna. The

alternative to an SCPA scheme is implementing the carrier combination stage at baseband

and using a single transmitter with a multi-carrier power amplifier (MCPA).

There are critical issues with the SCPA scheme. The precision of cavity filters becomes

an issue for small carrier spacing at frequencies in the PCS band. The final power

combination stage is nonlinear and lossy, resulting in further distortion. Although the

MCPA scheme relieves the designer from issues like RF combiner-loss, the combinedsignal at baseband has a time-varying envelope regardless of the digital modulation.

Thus, although the multi-carrier amplifier is a better option in terms of cost and

maintenance, it has a more strict linearity requirement.

Meeting these requirements becomes an even tougher problem, with wideband

modulation in individual carriers. With wideband signals, a significant amount of

hysteresis (Figure 2.8) is observed in a PA, showing signs of memory effects. These

effects are also known as frequency-dependent effects, since they have a pronounced

dependency on the instantaneous frequency of the signal as the bandwidth of modulation

increases. In order to minimize the overall distortion introduced by a power amplifier

operating on these signals, a detailed study of memory effects, along with the more

common non-linearity issue, is a must.


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Figure 2.7 - Two basic amplification schemes used in multi-carrier transmitters (RFPC

denotes RF power combination and BBPC stands for baseband power combination).

0 0.5 1 1.5 2 2.5 3 3.50

10

20

30

40

50

60

70

volts in

voltsout

Figure 2.8 - AM-AM curve for a 4-carrier W-CDMA signal showing hysterisis.


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2.4.1 Memory Effects

A system can be nonlinear, have memory, or be nonlinear with memory. A nonlinear

system generates new frequency components as seen in the previous section, while a

system with memory shapes existing frequency components. When an input signal does

not justify the narrowband assumption, the envelope characteristics at the output of a PA

do not remain static over the bandwidth of operation. These effects are visible as

hysteresis in the envelope transfer characteristic curves. The word hysteresis is used for

any residual effect, i.e., due to storage circuit elements in a PA the same input results in

different output, depending on the state of the PA. In other words, the circuit parameters

are unable to regain their original state after a transition. In order to understand the cause

behind these effects, we start off by classifying them as electrical memory effects, due to

frequency dependent envelope and node impedances in a PA, and thermal memory

effects, due to level-dependent temperature variation and variation in envelope and

component impedances resulting from this electro-thermal coupling.

In addition to the hysteresis, the presence of memory effects can be observed by

revisiting the two-tone test. As observed in the power series model for a PA, IM3

components on both sides of the fundamental frequencies is given by

3

3

3IM3

4a v= (2.6)

Now, (2.6) clearly states that the IM3 components are dependent on amplitude of the 3rd

order coefficient, but independent of tone spacing, 2 1 . However, the IM3 generated

in a typical PA does not behave in this manner. Figure 2.9 shows the actually measured

and polynomial-modeled phase of the lower IM3 component versus tone spacing. We can

observe the frequency-dependent variation at low and high modulation frequencies. This

observation points towards the existence of memory effects that a polynomial

nonlinearity model cannot depict.


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Figure 2.9 - Measured and modeled phase of IM3 as a function of tone spacing(Source: [Vuolevi01], Copyright IEEE, 2001. Used with permission.)

A discussion of the causes for electrical and thermal memory effects comes next. Since

we are specifically interested in wideband operation, emphasis will be on electrical

memory effects, followed by a brief outline for thermal effects. The above discussion, as

well as the following two sections, is based on [Vuolevi01].

2.4.1.1 Electrical memory effects

In order to develop an understanding of the causes for electrical memory effects, we need

to realize that in a transistor, there is more than a single nonlinearity mechanism. These

nonlinearities interact with each other producing additional IM components.

We consider a simple case of two, 3rd

order nonlinearities interacting in a transistor:

Nonlinearity between the input and gate junctions, and another between the gate and

output nodes, denoted by H and F respectively (Figure 2.10). H3 generates IM3

components at the gate that are amplified by F1, while H1 produces fundamental

frequencies used in the generation of IM3 components at F3. The second order

components at the output of H2 and the first order linear components at the output of H1

are combined in F2, creating IM3 at the output. Similar combinations result at othernodes. This simplified demonstration of interaction between nonlinearities in a PA

(Figure 2.11), indicates the fundamental limitation of the narrowband nonlinear model of

a PA. Figure 2.11 also reveals another observable artifact of memory effects, which is the

unequal distribution of IM3 components, or adjacent channel power, on either side of the


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19

band of interest (we will have more to say about this observation during the discussion on

PA modeling results).

Figure 2.10 Schematic showing the interaction of the two nonlinearities

(Source: [Vuolevi00], Copyright IEEE, 2000. Used with permission.)

The nonlinearities discussed above are usually generated by voltage dependency of the

gate and drain outputs on their respective node impedances, Zgg and Zdd . These nodeimpedances, in turn, depend on the internal and external impedances, where the external

impedances are a combination of the bias and matching impedances at the respective

nodes. Thus, nonlinearities are controlled by matching and bias networks, and their

design is critical for minimizing the generation of IM3 components. The impedances, or

reflection coefficients, a transistor wants to see looking at the source and load, are

matched to the actual source and load impedances using networks made from inductors

and capacitors. Typically, these networks are designed using single-tone measurements,

but they are accurate enough over the entire bandwidth of a PA, operating on relatively

narrowband signals. The problem occurs when wideband signals are used with very high

power amplifiers. Any variation in gate (base) impedance can cause significant electrical

memory effects. Also, bypass capacitors prove to be a significant source of memory


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effects. In order to limit memory effects, the envelope node impedances must be kept

very low or constant for the entire modulation bandwidth.

Figure 2.11 - The generation of IM3 from secondary mechanisms

(Source: [Vuolevi01], Copyright IEEE, 2001. Used with permission.)

2.4.1.2 Thermal memory effects

Thermal memory effects result from the electro-thermal coupling of the input and chip

temperature affecting low modulation frequencies (Figure 2.9). The thermal model of a

BJT is given in Figure 2.12, and the dissipated power for a BJT is given by

DISS CE CEp v i= (2.7)


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Figure 2.12 Thermal model of a BJT in (a) physical and (b) electrical configuration,

showing temperature at various levels(Source: [Vuolevi01], Copyright IEEE, 2001. Used with permission.)

Since two fundamental signals are being multiplied, the spectrum of the dissipated power

includes dc, second-order envelope, and second harmonics components. Due to the finite

mass of the chip, the thermal impedance acts like a low-pass filter rather than being

resistive in nature. Thus, the temperature variation of the chip is not instantaneous, and

frequency-dependent phase shifts exist. The chip temperature depends only on the dc and

envelope components of the dissipated power and the thermal impedances, due to the

low-pass nature of the impedance. Therefore, the temperature has the following simple

form.

2 2 2 2

2 2

( ) ( ) ( )

Where,

= Chip temperature

= Ambient temperature

= Thermal resistance of the chip

( ) = Freq

CHIP A TH DISS TH DISS

CHIP

A

TH

TH

T T R p dc Z p

T

T

R

Z

= + +

2 2

uency-dependent thermal impedance

( ), ( ) = Envelope and dc component of

the dissipated power

DISS DISSp p dc

(2.8)


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From (2.8) the fact that temperature variation is dependent on the frequency spectrum of

the signal, is evident. This signal-dependent variation in chip temperature affects

impedances and circuit components. Thus, thermal memory effects are unavoidable for

any temperature-dependent electrical parameters of the transistor such as gain, output

conductance, capacitance, etc. The generation of thermal memory effects is a two-step

process. Signal modulation variation changes the chip temperature. Chip temperature

variations result in changes in the electrical characteristics of circuit elements,

introducing distortion. Accordingly, the process is termed Thermal Power Feedback.

2.4.2 Characterization of Memory Effects

There is growing interest in the characterization of memory effects, and research efforts

at various levels are continually taking place. A two-tone test, although being a good test

for memory effects, is not capable of the detailed characterization that is needed to better

understand its causes. Thus schemes, using real modulated signals, are being developed.

The direct impact of PA distortion on these modulations can be monitored and analyzed.

This leads to recommendations for amplifier design and compensation techniques that are

customized for the specific signal format.

A two-tone test can be used to appreciate the presence of memory as outlined in

[Bosch89 and Vuolevi01]. The two-tone test cannot measure the memory effects due to

the IM components and thus [Vuolevi01] has modified the basic two-tone test to include

an IM distortion signal with two tones to measure the optimal input signal for canceling

IM memory effects. A more accurate method of characterizing memory effects is to drive

a PA with real modulated test signals, and analyze the resultant distortion. This

customized analysis can result in better strategies to counter PA distortion for the tested

signal environment. Boumaiza has presented a measurement setup (Figure 2.13) with the

capability of generating and characterizing PA distortion for arbitrary complex

modulation schemes, which is very useful for 3G signal formats and amplifiers

[Boumaiza02].

Memory effects have a negative impact on the degree of linearization possible through

the use of a specific linearization scheme (linearization schemes are dealt with in the next


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chapter). Therefore, it is necessary for PA designers to carefully calibrate the matching

networks in order to minimize these effects. As a case study, Bosch and Gatti [Bosch89]

have shown the reduction of memory effects using adjustment in bias impedances of a

20W bipolar amplifier. In terms of performance of a linearization scheme, memory

effects must be sufficiently suppressed, or cancelled, before any significant suppression

of IM3 components can be achieved.

Figure 2.13 - Measurement setup for characterizing 3G amplifiers.

2.5 Behavioral Modeling of Power Amplifiers

A behavioral model of a device is defined as numerical mapping between its input and

output, depicting a specific characteristic or behavior of the device. Modeling of devices

is very important for computer-aided design of systems. Such models are necessary for

conducting numerical simulations on systems, for which analysis using traditional paper-

and-pencil approach is at least, too tedious.

2.5.1 Envelope Simulation of Power Amplifiers

One of the established facts from the discussion in the previous section is that PA

distortion analysis is tractable only for very simple scenarios. As the envelope

frequencies increase, or in time-domain, as the envelope variations become increasingly

random and rapid, distortion worsens. The analysis in such a case becomes too tedious


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for the paper-and-pencil approach, and CAD tools need to be used. Therefore, the

existence of sound numerical models for these complex scenarios is very important for

analyzing distortion effects using powerful CAD tools. Agilents Advanced Design

System is one such CAD tool which uses the Harmonic Balance method to solve

nonlinear differential equations for the circuit model of an amplifier. However, a short

simulation run can typically take many hours. Therefore, for time-efficient simulations,

emphasis should be put on simple models that accurately depict behaviors useful for the

specific study being conducted. In other words, rather than using generalized complex

circuit models, task-specific models should be developed in order to reduce simulation

runtime.

A clear difference between envelope simulation of a linear and nonlinear subsystem, is in

the required sampling rate. For a linear system, the sampling rate, or simulation

bandwidth, can be set twice that of the envelope bandwidth, using the Nyquist criterion

for reasonably accurate studies (it may be set to more than twice the signal bandwidth in

order to account for numerical inaccuracies). However, since a nonlinear system

generates additional frequency components, or expands the envelope bandwidth of the

signal, the simulation bandwidth has to be much larger than that suggested by the Nyquist

criterion. The actual sampling rate to be used depends on the order of the nonlinearity and

the number of significant IM components.

2.5.2 Narrowband Amplifier Model

For a narrowband3

baseband signal envelope ( )bbx t , the complex output envelope of an

amplifier is given by

( ) ( )( ) ( ) ( ) exp ( )BB BB BB BBy t x t x t j x t = (2.9)

3 Here, and in the rest of the thesis, the word narrowband means that the signal satisfies the narrowband

criterion or assumption.


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Here, the functions ( ) and ( ) , respectively, represent AM-AM and AM-PM

nonlinearity. ( ) and ( ) are typically modeled using a Taylor or power series as

shown in (2.1). The modeling is based on single-tone measurements at the carrier

frequency since the envelope characteristics of a PA do not change considerably over the

spectrum. A Single-tone model is accurate enough for conducting studies on a variety of

modulations such as 4 -DQPSK, OQPSK, etc.

High power amplifiers operating on wideband modulation generate frequency-dependent

effects, which appear on the envelope characteristics of a PA as hysterisis loops. Simple

narrowband models, based on single-tone data, fail to capture envelope behavior for these

modulations. Therefore, a different strategy is required to model nonlinearity in the

presence of memory.

2.5.3 Models with Memory

Dynamic systems, or systems having memory, are the most common member of the

nonlinear class of systems. Almost all systems have delay associated with them and that

delay appears as phase shifts between input and output. Systems which show an

instantaneous phase variation, apart from a nonlinear gain pattern, are called quasi-

memoryless systems, and the single-tone model is well-suited to represent them. A

nonlinear system with memory, such as a PA operating on wideband signals, is most

comprehensively modeled using a Volterra series.

2.5.3.1 Volterra models

The Volterra series is commonly known as a Taylor series with memory. A third-order

Volterra series for a discrete-time system with ( )x n and ( )y n as its input and output

respectively, can be written as


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1 2 2

3 3 3

1 1 1(1) (2)

,

0 0 0

1 1 1 (3)

, ,

0 0 0

( ) ( ) ( ) ( )

( ) ( ) ( )

N N N

k k l

k k l

N N N

k l m

k l m

y n h x n k h x n k x n l

h x n k x n l x n m

= = =

= = =

= +

Here,(1)

kh ,(2 )

,k lh , and(3)

, ,k l mh are the first-, second- and third-order discrete Volterra kernels,

while 1 , 2N , and 3 are memory lengths for each order. The number of kernels

required to represent such a model is2 3

1 2 3N N+ + . Obviously, computational

complexity for even a simple third-order system is prohibitive. Thus, some of the

accuracy of the Volterra model can be traded off with simpler schemes.

2.5.3.2 Wiener models

A conceptually simpler method to model nonlinearity with memory, is to independently

model nonlinearity and memory. A linear filter can be used to model memory of the

system, followed by a quasi-memoryless nonlinearity using a narrowband polar model.

The accuracy of such a model depends on the amount of memory in the system and the

degree of its nonlinearity, and is reasonable only for a weakly nonlinear PA with short

memory.

Figure 2.14 - The Wiener structure for nonlinear systems with memory.

2.5.3.3 Nonlinear Tapped delay line (TDL) models

The narrowband model of the PA is accurately represented using a complex-gain

polynomial. A straightforward extension for systems with memory can be obtained by

using a time-delay structure, where the complex polynomials at each tap of the delay line

model the combined effect of memory and nonlinearity in the system, as depicted in

Figure 2.15. This model is in fact a truncated form of a Volterra series. Mathematically,

LINEAR

FILTER

AM-AM &

AM-PM

x(n) y(n)


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Figure 2.15 - Time-delay line model for a PA.

1 1

0,0 ,

0 1

( ) ( ) ( ) ( )M N

j

k j

k j

y n x n a x n k a x n k

= =

= + (2.10)

The delay between the taps can be adjusted to obtain better accuracy; one sample delay is

used under normal circumstances. [Kim01] shows results for using such a scheme for

modeling a PA operating on 1 and 3 UMTS carriers. The results show a markedimprovement in the mean-squared estimation error, when the PA is operating on a 3-

carrier signal.

2.5.4 Modeling Results for a Power Amplifier with Memory

In order to fully appreciate the need for models with memory, performance comparison

between memoryless models and models with memory is necessary. In this section, we

describe the data collection procedure along with the simulation methodology for

modeling of a PA with memory, and compare estimation results with memoryless

(narrowband polynomial) models and a TDL model with memory.


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2.5.4.1 Data collection setup

Instead of using direct measurements from a real power amplifier, circuit envelope

simulation on Agilents Advanced Design System

is used to generate data for

developing the models. A circuit model for a two-stage base-station amplifier operating

at 917.5MHz is used. The exciting signal is a multi-carrier W-CDMA, with 32 DPCH per

carrier with a spreading factor of 128, based on the Test Model One specification

[3GPP02]. The simulator is designed to enable selection of up to four carriers, added

together in baseband using exponential modulators and an ideal combiner (Figure 2.17).

The average envelope power at the input of the PA is normalized by the total number of

carriers and baseband sampling frequency.

2.5.4.2 Circuit Modeled and TDL-Modeled envelope characteristics

In this section, we have presented measured and modeled envelope characteristics for a 4-

carrier, 32 DPCH per carrier, W-CDMA downlink signal. The RF bandwidth of the

signal is around 20MHz.

The models have been estimated using the standard linear least squares (LS) method. The

basic algorithm formulation can be found in any standard signal processing text [Kay93]

or [Haykin01]. The output envelope magnitude (Figure 2.18) clearly shows the increasedaccuracy of the TDL model, compared with the polynomial case. The TDL model results

in a -25 dB normalized mean-squared envelope error (NMSE) as compared to the -22 dB

for the polynomial scheme. Note that NMSE is mathematically defined as,

2

1

2

1

( ) ( )

NMSE

( )

N

measured modeled

n

N

measured

n

y n y n

y n

=

=

=

The envelope transfer characteristic for amplitude nonlinearity is shown in Figure 2.19.

Since the polynomial model cannot model the hysterisis, we can see a reasonably close

match between the measured AM-AM and that modeled using a 4 tap, 10th

order TDL.

The power spectral density (Figure 2.20) shows a clear picture of the memory effects in


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the frequency domain. As mentioned during the discussion on memory effects, an artifact

of memory is unequal distribution of IM3 components on either side of the spectrum of

interest. For a polynomial nonlinearity, the IM3 components are equally-distributed

around the desired band, visible in the figure using line AB. The TDL model, however,

comes very close to the measured spectrum, apparent in the unbalanced IM3 distribution,

and a close overlap between the measured and TDL-modeled spectral components.

2.6 Summary

To fully realize the need for the linearization of an amplifier, a discussion of distortion

effects in a PA is necessary. Based on [Cripps99], we have presented a brief, but useful

numerical analysis of Inter modulation distortion (IMD) using a simple nonlinearity with

two-tone signal. Generation of IMD is the fundamental issue with nonlinear power

amplifiers. Third-order products are most critical in determining the end-to-end

performance. Another distortion phenomenon is the appearance of frequency-dependent

memory effects in a PA as the bandwidth of modulation grows. A brief section, covering

the causes for these effects is presented, followed by a discussion of behavioral modeling

methods for a nonlinear amplifier. Traditionally, for TWT amplifiers, a polynomial

function has been sufficient to model gain and phase nonlinearities, assuming that the

envelope bandwidth is very small compared with the carrier frequency, so that the PA

offers a flat gain to the entire input spectrum (known as the Narrowband Assumption).

However, the polynomial model becomes inaccurate for solid-state amplifiers operating

on wideband-envelope signals and showing memory. Models with memory, such as

Wiener, Tapped Delay Line (TDL), and Volterra schemes, provide increased modeling

accuracy, in that order. The chapter is concluded with model estimation results for a 4-

tap, 10th

order TDL model of a base-station amplifier operating on a four-carrier, 32-

DPCH-per-carrier, W-CDMA signal.


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WCDMA3G_TestModel1W6

ModelGainFactor=1DPCHNum=32ScramblingCodeIndex=0

TESTmodel 1

3GPP

ComplexExpC19

InitialRadians=0RadiansPerSample=pi/13.12/(Samples_per_Chip/16)

MpyCx2M5

RaisedCosineCxR5

SquareRoot=YESExcessBW=.22

SymbolInterval=Samples_per_ChipLength=16*Samples_per_ChipInterpolation=Samples_per_ChipDecimationPhase=0Decimation=1

GainCxG6Gain=1/sqrt(2)

ComplexExpC8

InitialRadians=0RadiansPerSample=-pi/13.12/(Samples_per_Chip/16)

MpyCx2M4

RaisedCosineCxR4

SquareRoot=YESExcessBW=.22SymbolInterval=Samples_per_ChipLength=16*Samples_per_ChipInterpolation=Samples_per_ChipDecimationPhase=0

Decimation=1

WCDMA3G_TestModel1W5

ModelGainFactor=1DPCHNum=32

ScramblingCodeIndex=0

TESTmodel 1

3GPP

BusMerge2B1

AddCxA2

GainCxG1Gain=sqrt(Samples_per_Chip)

Figure 2.16 Snapshot of the data-generating portion of the envelope simulator.


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Power Amplifier Circuit Model

Occupied Band Width

TimedToCxT1

TimedSink

Timed_Env_In

ControlSimulation=YES

Stop=DefaultTimeStop

Start=DefaultTimeStart

Plot=Rectangular

Test_TX

X1

IF_Freq=IF_Freq MHz

Stop_time=Tstopsample_period=Tstep*Samples_per_Chip/64

RF_Freq=RF_Freq MHz

SpectrumAnalyzer

Spec_Out

WindowConstant=0.0

Window=none


Start=0

Plot=Rectangular

GainRF

G3

GComp="0 0 0"

GCSat=1

PSat=1 W

dBc1out=1 W

TOIout=3 W

GCType=noneNoiseFigure=0

Gain=dbpolar(0,0)

CxToTimed

C10

FCarrier=IF_Freq MHz

TStep=(Tstep*Samples_per_Chip/64)sec

SpectrumAnalyzer

Spec_Probe1

WindowConstant=0.0

Window=none


Start=0

Plot=Rectangular

Figure 2.17 PA circuit model section of the envelope simulator.


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Figure 2.18 Comparison between output envelope for 4-carrier WCDMA.

Figure 2.19 - Measured and Modeled AM-AM characteristics (Polar method refers to the

memoryless polynomial method).


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Figure 2.20 - PSD comparison at the output of the PA for a 4-carrier WCDMA.


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3.1 Introduction

The most common method applied to minimize distortion by a nonlinear amplifier is to

avoid setting its operating point where the signal envelope fluctuates inside the

compression region. Input power backoff can be used to force the amplifier to operate in

the linear region and therefore, is a primitive form of linearization. However, input back-

off reduces the maximum amount of power, available at the output of an amplifier and

thus is not a feasible solution for most applications. In such a case, a more efficient

linearization scheme is required. In simple words, Linearization is a process which

enables linear amplification of a signal in the presence of nonlinear components, by

canceling the distortion introduced by them. Linearization schemes can be broadly

classified as open-loop and closed-loop schemes. A closed loop technique generally

results in better linearity, but it has limited correction bandwidth (defined as the

bandwidth over which the linearization method can reduce distortion) due to the finitefeedback loop gain and, in addition, suffers from stability issues. On the other hand, an

open-loop technique results in slightly worse performance, but it has good correction

bandwidth and is inherently stable due to the absence of feedback. In addition to

Linearization schemes, there are amplifier architectures that avoid distortion. These

linear transmitters have very good theoretical efficiencies, but they have

implementation issues at RF.

The following sections briefly discuss common linearization schemes covered in the

literature. Comments on their shortcomings in the presence of frequency-dependent PA

distortion, is included.


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3.2 Canceling Nonlinear Distortion

Only a few of the schemes mentioned in this section are suitable for a complete digital

implementation. However, a basic understanding of all is the key to a proper

understanding of the subject of digital linearization.

3.2.1 Cartesian Feedback Method

The basic Cartesian Feedback scheme is outlined in Figure 3.1 [Cripps99]. The Inphase

and Quadrature signals, ( )I t and ( )Q t , are filtered symbol sequences at baseband. The

signals are fed through differential amplifiers into an I-Q modulator which modulates an

RF carrier of frequency cf to formulate the signal ( )S t , where

( ) ( ) cos ( )sinc cS t I t w t Q t w t = + (3.1)

Figure 3.1 - Cartesian feedback transmitter.

The signal ( )S t is then fed to a PA. A small portion of PA output is coupled to the

downconverter, which generates a distorted version of the Inphase and Quadrature

symbol sequence at its output. The sequences are directly compared with the input


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symbol sequence in the differential amplifiers generating the loop error. This loop error is

then filtered to remove high frequency contents and fed to the PA. Due to the real-time

adaptive nature of the scheme, it is particularly suitable to various drifts in PA

characteristics due to aging and heating. Sufficiently high loop gain ensures stable

operation and excellent reduction in distortion, over narrow bandwidths.

There are several limitations in this scheme. Although the correction precision is very

good, performance is tied to the gain of the feedback loop, the bandwidth of the

differential amplifiers, carrier drift, and most importantly, the linearity of the

downconverter and demodulators.4

Phase adjustments are required as shown in Figure 3.1

to cancel imbalances and leakage in the modulators and demodulators. A DC gain offset

is another detrimental by-product of the down-conversion stage and operational

amplifiers, providing feedback gain. In this regard [Boloorian97] has proposed new

methods to automate these adjustments and corrections.

This scheme can be modified for digital implementations by applying the correction at

baseband. However, the presence of memory can significantly reduce its effectiveness

and the region of stable operation owing to its memoryless nature. Also this scheme is not

suitable enough to handle wideband signals due to its high loop gain, which is necessary

for stable operation.

3.2.2 Feedforward Amplifier

A feedforward amplification scheme cancels distortion by, in essence, isolating the

distortion and subtracting it from the output. However, although simple in theory, there

are several issues and complications with this scheme. Nonetheless, the feedforward

scheme has been, and still is, the most popular linearization technique.

4 The linearity of the downconverters and demodulators is critical for all schemes utilizing any form of

feedback.


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Figure 3.2 shows the basic scheme of a feedforward amplifier. The output of the power

amplifier is suitably attenuated before being combined with the reference, or input signal,

in C1. The error signal, at the output of C1 is amplified using an error amplifier, and

combined with the power amplifier output at C2; C1 and C2 are used as differential

combiners.

Figure 3.2 - Block diagram of a Feedforward system

For a typical 10-dB coupling ratio at C2, 90% of the power from the main PA reached the

load, while only 10% reaches from the error amplifier. The error amplifier, therefore,

must produce ten times the power of the distortion in the main amplifier. This large

difference in the PAPR between the desired and error signal, makes the error

amplification process much less efficient than main power amplification. In addition, to

enable reasonable correction, it may be necessary to operate one or both amplifiers well

into backoff to improve linearity. The overall average efficiency of a feedforward

transmitter may, therefore, be only 10%15% for typical multicarrier signals [Raab02].

There are several other sensitive issues that need to be addressed. A difference in aging

and temperature characteristics of the two amplifiers results in a mismatched output, and

C1

C2

T1

T2


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consequently, distortion due to an out-of-phase signal combination. However, with

proper calibration, the sensitivity can be minimized. Furthermore, accuracy of the gain

and phase tracking circuits in the two paths is critical for proper isolation and cancellation

of distortion. The system is also sensitive to path delays, T1 and T2, which are inserted

for time synchronization. A delay mismatch is magnified in the error amplifier, and may

introduce more distortion as a result. Parsons quantifies the effect of shortening or

removing these delay elements on the overall achievable correction [Parsons94]. As

outlined in the previous paragraph, power efficiency of the system is a concern given the

fact that the error signal, at the output of C1, needs to be small for a check on the size of

the error amplifier.

Several schemes are used to track and compensate for delay amplifier mismatch. The

signal cancellation loop can be optimized by measuring and minimizing error signal,

while the distortion cancellation loop can be tuned by minimizing out-of-band power

[Sundstrom95]. Pilot-based schemes have been suggested where the level of the pilot

signal can be used to tune the error and distortion cancellation loops [Narahashi91]. Since

adaptive amplitude and phase matching is needed due to component aging, temperature

drifts, and changes in the operating frequency, several gradient-based adaptation schemes

have been presented in the literature for gain and phase tracking and have been

implemented on a DSP [Grant96, Chen00]. A complete analog adaptive implementation

for the gain-phase tracking block is suggested by Zozaya, combining simplicity with

robustness [Zozaya01].

For wideband input and error signals, the main and error amplifier may enhance or

reduce the memory effects, depending on the power combination process at C1 and C2.

Research is required in this area to reach an answer.


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3.2.3 Predistortion

The working principle of a predistorter (PD), shown in Figure 3.3, can be stated in

following form [Muhonen01].

The input to the transmitter, inr , is mapped to the output of a PD,

,pd outr , in such a way that ,pd outr , when operated with a nonlinear

amplifier, produces a linear multiple of inr .

Predistortion is a nonlinear mapping procedure applied before the power amplifier that

attempts to produce equal and inverse IM components to those generated in a PA. The

scheme, being an open-loop technique in principle, has worse performance in comparison

to a cartesian feedback linearizer, but does not have stability problems, and has a much

wider correction bandwidth, which is desirable for current and future modulation formats.

Figure 3.3 A basic predistortion transmitter.

Before we present the several categories of predistorters, it is necessary to understand

specific tradeoffs in its operation. Figure 3.4 illustrates a nonlinear and desired linear

AM-AM envelope characteristics of a PA. Theoretically speaking, since an amplifier

cannot operate above its rated value, the dynamic range of input power for linear

operation (RLO or range of linear operation), decreases as desired linear gain is

increased. Curve-1 shows the situation where the desired linear gain is equal to the gain

of linear region of an amplifier, while Curve-2 has a lower gain setting and evidently, an

increased RLO. Thus, there is a tradeoff between the desired power gain and RLO. Given

a certain performance specification, one method to find optimal value for the desired gain

inr ,pa out inr K r= ,pd outr


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is to use input PAPR, i.e. a reduction in linear gain setting, for increasing input PAPR.

However, a better and more efficient method to achieve this tradeoff, is to use envelope

distribution for gain adjustment. Balister has shown that the PAPR value of a signal

envelope can provide a very conservative measure of its time-variability, and does not

provide a complete picture [Balister01]. The envelope distribution, in addition to PAPR

value, must be used for making a more accurate observation of the signal behavior. Thus,

the RLO can be increased above the conservative limit set by the PAPR method, based on

the observation that an envelope may, in fact, only spend a considerably small amount of

time near its peak value, resulting in only a fractional increase in distortion.

Figure 3.4 - Relationship between RLO and linear gain of a general predistorted system.

(Range-1 and Range-2 represent RLO for curve-1 and -2, respectively.)

There can be several ways in which a predistorter can be implemented. We briefly

mention important classes of predistorters.


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Predistortion for Nonlinear PA with memory

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